The Veronica Mars TV series ended with unfinished mysteries and unanswered questions.

The Veronica Mars movie takes place 10 years after the end of the TV series. During those 10 years, Veronica attended Stanford for a Psychology degree, then went to Columbia for a Law degree, and is currently a lawyer in New York. During her interview at Truman, she reveals she did not work a case after she moved to Stanford.

Did Rob Thomas or any other show creator reveal any additional information for the missing 10 years during interviews? An extended conversation would be welcome or even a link to a video.

Specific questions:

Did Veronica get an internship at the FBI? What happened to that, and why didn't she join the FBI?

The cast is very cognizant of the fact that the film would not have
been made without the money donated by the fans, and Thomas said that
once he knew it'd be financed that way, his original idea, which
involved Veronica as an FBI agent, would no longer work. "I sort of
abandoned that FBI idea and tried to find a plot to get back to all
the characters we know and love," said Thomas. "Because if the fans
are going to pay for a movie, I want them to see the characters they
love."

In other words, if Veronica were a rookie FBI agent, she would obviously be in a different geographical location, well away from all the former characters. To prevent this possibility, the movie takes place primarily over an old high school reunion so many of the characters can be reunited.

I could see how [the FBI idea] would have worked as a successful season of
Veronica Mars, and definitely one that I, and plenty of fans, would
watch and enjoy. It was a smart move to reset the setting and
characters, because that would make the show more accessible to new
viewers — which is exactly what The CW was looking for. The FBI angle
turns it into more of a crime drama with a whip-smart lead, rather
than a teen drama with a sassy lead (because, unfortunately, teen
dramas will always get a bad rep but crime shows will always be
popular).

In fact, that’s where the Season 4 pitch wins over the movie: Veronica
as an FBI agent makes more sense than Veronica as a lawyer. And sure,
the movie addresses the former career option and, yes, Veronica would
make a hell of a lawyer (remember the “One Angry Veronica” episode?),
but it’s not her. She needs to be constantly on the move, solving
cases, and in the mud. Thomas seems to agree; a few months prior to
starting the Kickstarter, he was still intent on having the film
center on Veronica in the FBI. Once the show graduated from high
school, it would naturally become more about the mysteries than the
setting.

But the Veronica Mars movie was fan service, and that’s why it worked
so well. The fans are the reason it happened, so Thomas had to abandon
his original idea because there was no way to do it while bringing
back all of the favorites from Neptune. The movie was packed with
callbacks and inside jokes — there is even a clever exchange about
this pilot pitch, referencing Veronica’s FBI career as something that
happened in another life — because, essentially, that’s what fans paid
for. The movie wasn’t much more than a long episode of the show, but
that’s exactly what I loved about it. Yet I also loved where the show
was going to go in the season-that-never-happened. It would have been
great to see a longer version of Veronica adjusting to the FBI and
generally just kicking ass and shooting bad guys. And maybe we’ll get
some stories of her in the law enforcement eventually (there are
rumors of a second movie, and Thomas hasn’t been shy about his desire
for a Netflix deal) but for now, Veronica’s intriguing alternate life
will remain too brief.

I think the logic applied here is completely rational. It just wouldn't have worked to have many of the old cast back and Veronica in the FBI.

Who filmed the sex tape?

The Washington Post did a recap of the series finale before the movie, commenting:

The series finale seemed more like a season finale. It was
unsatisfactorily open-ended and left many unresolved issues. Last we
saw, Veronica she was a freshman studying criminology at Hearst
College in California. A sex tape of her and boyfriend Stosh “Piz”
Piznarski (Chris Lowell) mysteriously surfaced. Neptune’s resident bad
boy and Veronica’s ex-boyfriend Logan Echolls (Jason Dohring) assumed
Piz was the culprit and beat him to a pulp.

Once it was revealed that
a secret collegiate society (modeled after Yale’s Skull and Bones) was
actually behind the tape, Veronica became furious at Logan. She tells
Logan their friendship is over and wants nothing to do with him, thus
seemingly choosing the sweet, levelheaded Piz over her former,
long-time flame (continuing an epic love triangle that picks up in the
movie).

Vengeful, Veronica hunts down the secret society and stages a
break-in at a billionaire’s mansion to collect evidence on the
association’s members. While escaping the residence, she is caught by
security cameras. Her father, former private eye and current sheriff
Keith Mars, protects her identity by destroying the surveillance
footage and, by doing so, jeopardizes his re-election campaign. The
final scene shows Veronica casting a ballot for her dad in the sheriff
election and walking off alone, seemingly debating whether her
sleuthing has caused more harm than good.

I never finished the series myself, but this sounds like that thread was fairly well resolved in the TV series. I'm not aware of any mention of it in the movie.

Note: Rob Thomas tweeted from his official Twitter (@RobThomas) that
Keith Mars lost the Sheriff election to Vinny Van Lowe. He also had
plans for The Castle and Gory Sorokin. As for Logan and Veronica, he
refused to answer the question. As for the movie, he said “Don’t give
up.”

Unfortunately I can't find the direct tweet in question, but as I've seen this claim in a few places I assume it is true.

“She transferred away from Hearst college after that freshman year and
went up to Stanford and finished her undergraduate degree. And then
she went to Columbia Law School. She’s certainly been back to Neptune
from time to time to see her dad and Mac and Wallace, but she hasn’t
worked a PI case since the last one we all saw in the final episode of
season 3. She ruined her dad’s chance for re-election and she decided
that that path had caused too much chaos in her life.

“And so she hasn’t worked a PI case in nine years. And when we meet
her, she’s sort of like where Tom Cruise was in the beginning of The
Firm. She’s between having graduated and taken the bar and she’s
interviewing at various law firms in New York.”

Final Thoughts

In the same interview as the last quotation, the following exchange occurred:

Will the movie be new audience-friendly? According to Thomas:

“I want to avoid getting deep into the ‘Veronica Mars’ mythology.
We’re not going to get ghost visits from Lilly, as much as I joked
about having Amanda Seyfried in it. We’re not going to get into
Veronica’s backlog of cases. I want to make the movie accessible for
people who haven’t watched ‘Veronica Mars’ before. All you need to
know is that Veronica was a teenage private eye and she’s given it up.
And there will be a moment early in the movie that will bring her back
to Neptune and make her give being a private detective one more shot.”

Therefore, it does seem like Rob Thomas was trying to target new viewers as well as the old fans. It definitely appears logical then that some of the loose ends were simply left, as it would be too convoluted for new viewers to understand them. Other threads were resolved in the movie and most other discussion points appear to have been addressed in interviews with Rob Thomas and the cast.

Excellent. Just what i was looking for. Btw, did the why part of sex tape resolved? Not even addressed i feel. Anyway, enjoyed the movie & hope for a second one
– KharoBangdoMar 24 '14 at 12:06

@KharoBangdo: I think the why was as simple as wanting to screw with her (by ... er... watching her screw). But there were definitely a lot of loose ends in the show.
– Andrew MartinMar 24 '14 at 12:11

Did Veronica get an internship at the FBI? What happened to that, and why didn't she join the FBI?

This is addressed in the course of the film as an off-hand comment by one of the characters speaking to Veronica, summarized as, "Someone told me you joined the FBI.", which Veronica dismisses outright as a false rumor. This was an intentional nod to the proposed FBI season.